Breakpoint graph analysis is a key algorithmic technique in studies of genome rearrangements. However, breakpoint graphs are defined only for genomes without duplicated genes, thus limiting their applications in rearrangement analysis. We discuss a connection between the breakpoint graphs and de Bruijn graphs that leads to a generalization of the notion of breakpoint graph for genomes with duplicated genes. We further use the generalized breakpoint graphs to study the genome halving problem (first introduced and solved by Nadia El-Mabrouk and David Sankoff). The El-Mabrouk-Sankoff algorithm is rather complex, and, in this paper, we present an alternative approach that is based on generalized breakpoint graphs. The generalized breakpoint graphs make the El-Mabrouk-Sankoff result more transparent and promise to be useful in future studies of genome rearrangements